Find shortest primary decomposition.

Question

Let $A=k[x,y,z]$ and let $T_1=(x,y)$, $T_2=(x,z)$. Define $I=T_1T_2$ and calculate the shortest primary decomposition of $I$.

I dont know where to start and I am looking for hints, how should I think when I want to find a primary decomposition? I found myself just guessing, trying to find a decomposition and hoping it was primary and of course it did not work out.

Answer 1

We have $I=(x^2,xy,xz,yz)$. Using the method from this answer we get $$I=(x,y)\cap(x,z)\cap(x^2,y,z),$$ a minimal primary decomposition.